I would like to know the reason why the equation (14) in the paper by Yamada is called the Toda equation.
\begin{equation}
\left[\frac12\sum_{i=1}^N\left(y_i\frac{\partial}{\partial y_i}-y_{i+1}\frac{\partial}{\partial y_{i+1}} \right)^2+\sum_{i=1}^N u_iy_i\frac{\partial}{\partial y_i}+{\rm const}\sum_{i=1}^N y_i\right]Z=0.
\end{equation}

This equation was obtained by taking the decoupling limit of the equation (12) which is the equation conformal blocks in $SL(N)$ WZNW theory satisfy.

Does a certain correlation function in Toda conformal field theory satisfy the equation (14)? If so, there should be a realization of a full surface operator by M2-branes in the context of the AGT relation.

What kind of correlation functions in Toda conformal field theory satisfy the equation (14)?